Globally optimal spatial estimation in robotics via trace-constrained semi-definite programming
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Abstract
Many problems in robotics involve estimating spatial quantities such as rigid-body rotations and transformations. Because these variables lie on nonlinear manifolds, the resulting optimization problems are inherently nonconvex, making it difficult to obtain globally optimal solutions. Recent advances in semi-definite relaxations have shown promise in achieving globally and certifiably optimal solutions; however, their practical performance critically depends on recovering rank-1 solutions from relaxed formulations, which remains a challenging task. This dissertation presents an efficient and systematic framework for modeling, relaxing, and solving problems in robot kinematics, estimation, and calibration using trace-constrained semi-definite programs (SDPs). In this formulation, the decision variables are positive semi-definite (PSD) matrices with fixed trace values, enabling gradient-based refinement of relaxed solutions toward rank-1 optimal candidates. The contributions are threefold. First, we propose customized trace-constrained SDP relaxations for common spatial quantities in robotics, including rotations and translations. Second, we investigate the topology of the relaxed convex domain and develop strategies to efficiently obtain rank-1, low-cost solutions. Third, we introduce a modular virtual-robot abstraction that unifies the modeling of diverse robotic problems, including inverse kinematics, perspective-$n$-point estimation, and robot calibration.
Description
2026
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Attribution 4.0 International